Maximum flow and topological structure of complex networks

TL;DR

This paper investigates maximum flow between node pairs in complex networks, revealing relationships with degree distribution and edge-biconnected components, and applies findings to real-world Internet data.

Contribution

It introduces a novel link between maximum flow statistics and network topological features like edge-biconnected components in complex networks.

Findings

Average maximum flow scales linearly with minimum node degree

Disjoint paths are distributed within edge-biconnected components

Results are validated on Internet autonomous system network data

Abstract

The problem of sending the maximum amount of flow $q$ between two arbitrary nodes $s$ and $t$ of complex networks along links with unit capacity is studied, which is equivalent to determining the number of link-disjoint paths between $s$ and $t$. The average of $q$ over all node pairs with smaller degree $k_{\rm min}$ is $<q>_{k_{\rm min}} \simeq c k_{\rm min}$ for large $k_{\rm min}$ with $c$ a constant implying that the statistics of $q$ is related to the degree distribution of the network. The disjoint paths between hub nodes are found to be distributed among the links belonging to the same edge-biconnected component, and $q$ can be estimated by the number of pairs of edge-biconnected links incident to the start and terminal node. The relative size of the giant edge-biconnected component of a network approximates to the coefficient $c$. The applicability of our results to real world networks is tested for the Internet at the autonomous system level.